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<p>Numerical approximation of quantum states via convex combinations of states with positive partial transposes (bi-PPT state) in multipartite systems constitutes a fundamental challenge in quantum information science. We reformulate this problem as a linearly constrained optimization problem. An approximate model is constructed through an auxiliary variable and a suitable penalty parameter, balancing constraint violation and approximation error. To solve the approximate model, we design a linearized proximal alternating direction method of multipliers (LPADMM), proving its convergence under a prescribed inequality condition on regularization parameters. The algorithm achieves an iteration complexity of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(1/\epsilon ^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>ϵ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for attaining <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-stationary solutions. Numerical validation on diverse quantum systems, including three-qubit W/GHZ states and five-partite GHZ and multiGHZ states with noises, confirms high-quality bi-PPT approximations and decomposability certification, demonstrating the utility of our method for quantum information applications.</p>

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Approximating quantum states with positive partial transposes in multipartite system via linearized proximal alternative direction method of multipliers

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Jingwen Fan,
  • Deren Han,
  • Lin Chen

摘要

Numerical approximation of quantum states via convex combinations of states with positive partial transposes (bi-PPT state) in multipartite systems constitutes a fundamental challenge in quantum information science. We reformulate this problem as a linearly constrained optimization problem. An approximate model is constructed through an auxiliary variable and a suitable penalty parameter, balancing constraint violation and approximation error. To solve the approximate model, we design a linearized proximal alternating direction method of multipliers (LPADMM), proving its convergence under a prescribed inequality condition on regularization parameters. The algorithm achieves an iteration complexity of \(O(1/\epsilon ^2)\) O ( 1 / ϵ 2 ) for attaining \(\epsilon \) ϵ -stationary solutions. Numerical validation on diverse quantum systems, including three-qubit W/GHZ states and five-partite GHZ and multiGHZ states with noises, confirms high-quality bi-PPT approximations and decomposability certification, demonstrating the utility of our method for quantum information applications.